where Sn⁢(x)subscriptSnxS_{n}(x) is the nthsuperscriptnthn^{\mathrm{th}}partial sum of (1) (n= 1, 2,…n 1 2normal-…n\,=\,1,\,2,\,\ldots). The uniform convergenceimplies the existence of a number nεsubscriptnεn_{\varepsilon} such that on the whole interval we have

Let now n>nεnsubscriptnεn>n_{\varepsilon} and x0,x0+h∈[a,b]subscriptx0subscriptx0habx_{0},\,x_{0}\!+\!h\in[a,\,b] with h>0h0h>0. Since every fn⁢(x)subscriptfnxf_{n}(x) is continuous from the right in x0subscriptx0x_{0}, the same is true for the finitesumSn⁢(x)subscriptSnxS_{n}(x), and therefore there exists a number δεsubscriptδε\delta_{\varepsilon} such that